PropositionStatement: AI-adaptedProof: AI-adaptedPipeline-generatedaudited 2026-09-12

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Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
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Shifting a filtered complex reindexes its spectral sequence

Statement

For integers a,b define $D_{n} = C_{n - a}$ , $d_{D} = d_{C}$ , and $G_{p} D_{n} = F_{p - b} C_{n - a}$ . Then $E_{p, q}^{r} (D, G) ≅ E_{p - b, q + b - a}^{r} (C, F)$ canonically, with equality when the same quotient models are used. This identification commutes with differentials and next-page maps and identifies the finite filtered abutments. This is a sign-free degree translation, not the signed triangulated shift.

Facts & Assumptions

Given: A filtered chain complex and two integers a,b, with D and G as defined in the statement.

[F1]

The next-page map is a natural isomorphism induced by the next-cycle inclusion (The next page is the homology of the current page).

[F2]

Maps preserving the filtered construction induce its canonical quotient maps (A filtered chain map induces a morphism of spectral sequences).

[F3]

Finite filtered convergence identifies the stable quotient with image-filtered homology (Bounded filtered complex spectral sequence abuts to filtered homology).

[F4]

The page formulas include both numerator and denominator bounds (R page of the spectral sequence of a filtered complex).

Proof

technique · direct

1.1

Put n=p+q. The reindexed total degree is $(p - b) + (q + b - a) = n - a$ . Direct substitution gives $A_{p, n}^{r} (D, G) = A_{p - b, n - a}^{r} (C, F)$ . The lower denominator becomes $A_{p - b - 1, n - a}^{r - 1}$ and the boundary denominator becomes $d (A_{p - b + r - 1, n - a + 1}^{r - 1})$ . For r=0 the quotient is $F_{p - b} C_{n - a} / F_{p - b - 1} C_{n - a}$ . Thus the quotients in [F1] agree under these indices.

F1algebraF4

2.1

The differential is still $d_{C}$ , and the target index substitution is $(p - r - b, q + r - 1 + b - a)$ on either route. The inclusions and quotient maps defining α in [F1] also coincide under substitution; hence their induced isomorphisms commute. This is the same uniqueness of quotient descent used in [F2]. No sign is introduced because $d_{D}$ was explicitly defined to equal $d_{C}$ .

F1F2step 1.1algebra

3.1

The cycle and boundary objects of $D_{n}$ are those of $C_{n - a}$ , so $H_{n} (D) = H_{n - a} (C)$ with $G_{p} H_{n} (D) = F_{p - b} H_{n - a} (C)$ by the image definition. Finite bounds are translated by b in filtration and a in degree. Applying [F3] gives precisely the same index relation on stable graded quotients.

F3step 1.1step 2.1∎

Source notes

Weibel, Chapter 5, Construction 5.4.6 and Lemma 5.4.7, pp.133–134; Sharifi, Theorem 4.2.3, pp.91–92. Increasing homological indices are used here. The sign-free degree/filtration translation here is distinguished from the décalage of Weibel Exercise 5.4.3.

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Sources

Charles A. Weibel, An Introduction to Homological Algebra, Chapter 5 (standard reference, not scraped)
Romyar Sharifi, Homological Algebra (standard reference, not scraped)